318]
RA.NDOM. DISTRIBUTION OF LUMINOUS SOURCES
'579
lat although cos°|^ appears in the denominator of (33), it is compensated heir cos |^ = 0 by a similar factor in the numerator. In the integration ith respect to ^
sin Y^Y = — 4 cos i v. d (cos t y).
/V /v « /V \ ij /V'
r we write ^ for 2^^005 %%, the mean sought may be written
(sin -dr — \lr COS -vlr)2 •
i-----1—T------I__L. dty .................-....(37)
'UT° v '
10 range for -^ being from 0 to 2kR. The integration can be effected by parts." .. "We have
<» f*i5 • A,* f*,4 ^ '
is small, the expression on the right becomes
18'
> that the integral between 0 and ^ is itensity is
9n2 2*\Jr sin -fy cos ty — sin2 ^ — •xp + ^*
simply. In general, the mean '"'( }
i which stands for ZkR.
That the intensity, whether in one direction or in the mean of all directions, tiould be proportional to n2 is, of course, what was to be expected. And, since he effect is here a surface effect, it maybe identified with the ordinary surface 3flexion which occurs at a sudden transition between two media of slightly iffering refrangibilities, and is proportional to the square' of that difference. f, as in a former problem, we suppose the discontinuity of the transition to e eased off", this renexion may be attenuated to any extent until finally there 3 no dispersed wave at all*.
When we pass from the continuous uniform distribution to the random istribution of n discrete and very small obstacles, the term in n" representing eflexion from the surface remains, and is now supplemented by the term in n, .ue to" irregular distribution in the interior. It is the latter part only with rhich we are concerned in a question such as that of the blue of the sky.
It must never be forgotten that it is the "expectation" of intensity which 3 proved to be??. In any particular arrangement of particles the intensity my be anything from 0 to w2. But in the application to a. gas dispersing i.ght; the motion of the particles ensures that a random redistribution of liases takes place any number of times during an interval' of time less than ny which the eye could appreciate, so that in ordinary observation we are oncerned only with what is called the expectation.
* Gonf. Proc. Lond. Math. Soc. Vol. xi. p. 51 (1880) ; Scientific Papers, -Vol. i. p. 460.
37—2e result by 4?r. It may be remarked